Research Experience for Undergraduates

Content

The IMA summer REU gives students an experience in working on an interdisciplinary project involving mathematics. Students will work in teams of three with a faculty advisor and a postdoctoral mentor. This is a hands-on experience. There will be a few formal lectures. However, students will spend most of their time doing actual research. Students are expected to produce a poster and a publication-quality written report, and to give an oral presentation by the end of the 5-week period.

Reading and supporting material will be provided to successful applicants prior to the start of the program.

Deadline March 31, 2010. Applications are invited from students who will be entering Junior and Senior years in the Fall of 2010. There is no citizenship requirement. Only 9 students will be admitted to the program.

Stipend and allowance:

00 stipend, campus housing and meals, travel allowance of up to 0 each. Participants also have a 0 allowance for travel to a national meeting during the 2010-2011 academic year to present results of their REU work.

Requirements:

Must be full-time undergraduate mathematics major and must devote full time to the program and may not engage in other course work or employment during the 5 week period.

Problem Poser: Volkan Isler, Department of Computer Science and Engineering, University of Minnesota

Project description

In a pursuit-evasion game, one or more pursuers try to capture an evader who in turn tries to avoid capture. There are many variants of pursuit evasion games based on the environment (e.g. a polygon, graph), information available to the players (e.g. can they see each other at all times?), motion constraints (e.g. a car chasing an evader can not turn arbitrarily) and the definition of capture (in some games, the pursuer captures the evader if the distance between them is less than a threshold. In other games, the pursuers must see or surround the evader in order to capture it.)

We will study two fundamental pursuit-evasion games. The cops-and-robbers game takes place on a graph. At each turn, the players move along the edges. The evader (robber) is captured when one of the cops moves onto his current location. The lion-and-man game is a geometric version of the cops and robbers game. In the original version, the game takes place inside a circular arena. The players have the same maximum speed. The objective of the lions (pursuers) is to capture the man by moving onto the man's current location.

Research Questions:

Given a planar graph G, what is the number of cops necessary and sufficient to capture the robber on G? (the answer should be 1,2 or 3)

Given a polygon with obstacles, what is the number of lions necessary to capture the man?

In contrast to rigid boundaries, flexible solid boundaries can deform under the action of shear and normal stresses, resulting in the creation of surface waves. If the stresses are exerted by an adjacent flowing fluid, these waves may lead to a complicated, time-dependent flow. Important consequences of this modified flow include the alteration of mass and heat transfer rates and alteration of the stresses exerted on the solid surface. Such elastohydrodynamic instabilities, if better understood, could find application in a variety of areas including microfluidic mixers, membrane separations, and the rheology of complex fluids that undergo flow-induced gelation.

The schematic below shows a liquid flowing past a gel, a type of deformable solid. The liquid flow may be driven by a combination of boundary motion and externally applied pressure gradients. In the situation pictured, the flexible boundary is the interface between the liquid and gel. At a critical liquid flow rate, the initially flat liquid-gel interface becomes unstable, leading to a state in which waves travel along the interface. As a consequence, the liquid flow, which initially had parallel streamlines, becomes more complicated. This instability occurs even when inertia is completely absent; it is purely a consequence of having a deformable boundary.

Whereas there has been much theoretical work concerning the linear aspects of this instability, relatively little is known about its nonlinear aspects. For systems with fluid-fluid interfaces, it is known that one effective way of understanding nonlinear aspects of instability is the development and analysis of long-wave equations. These equations are essentially the leading order problem in an asymptotic expansion of the full governing equations, where the expansion parameter (assumed small) is the ratio of a characteristic vertical distance to the instability wavelength. The goal of this project is to derive and analyze long-wave equations for the system shown in the above schematic. After the equations have been derived, it will be of interest to perform a linear stability analysis, a weakly nonlinear analysis, and direct numerical simulations. It will also be of interest to compare the linear stability analysis results with the results of a similar analysis of the full governing equations to determine how well the long-wave model captures the linear aspects of the instability.

References

The linear aspects of the elastohydrodynamic instability described above are discussed in:

The approach taken in this paper can be adapted to the problem described above if the liquid-air interface is replaced by a rigid solid boundary. As a first step, it would be worthwhile to (i) completely neglect inertia, (ii) assume that a linear constitutive model for the gel is appropriate, and to (iii) suppose that a long-wave description is appropriate. The reason for (i) is that the instability is known to occur in the absence of inertia. The reason for (ii) and (iii) is that linear models and long-wave descriptions sometimes work surprisingly well outside of the regimes in which they are strictly valid.

Efficient processing and analysis of massive and high-dimensional data requires its reduction by a simpler model. The simplest and most common geometric data modeling uses a single affine subspace, but it does not represent well many types of data. The next one uses a combination of several affine subspaces and is often referred to as Hybrid Linear Modeling (HLM). Various HLM algorithms have been suggested and applied in diverse problems such as segmenting motions in video sequences and clustering faces under varying illuminating conditions. In this project we will review all algorithms for hybrid linear modeling we know of and carefully compare them on artificial and real data, while trying to explain their successes and failures and possibly suggest improvements or effective combination of several algorithms.